Lattice.
A lattice \(\Lambda \)  is a discrete additive subgroup of \(\mathbb {R}^n\) .
For a rank-\(n\)  lattice in \(\mathbb {R}^n\) , the rows of a \(n\times n\)  generator matrix \(G\)  constitute
a basis of \(\Lambda \)  and any lattice point \(x\)  is obtained via \(x=z \cdot G\) , where \(z \in \mathbb {Z}^n\) .
The Gram matrix is \(\Gamma =G\cdot G^T=(GQ) \cdot (GQ)^T\) ,
where \(Q\)  is any \(n \times n\)  orthogonal matrix.
All bases defined by a Gram matrix are equivalent modulo rotations and reflections.
A lower triangular generator matrix is obtained from the Gram matrix
by Cholesky decomposition [1]}.
For a given basis \(\mathcal {B}=\lbrace  g_i \rbrace _{i=1}^n\)  forming the rows of \(G\) ,
the fundamental parallelotope of \(\Lambda \)  is defined by
\(\mathcal {P}(\mathcal {B}) = \lbrace  y \in \mathbb {R}^n : y=\sum _{i=1}^{n}\alpha _{i}g_{i}, \ 0 \le \alpha _{i} < 1 \rbrace .\) 
